wave propagation through arrays of unevenly spaced vertical piles

Wave propagation through arrays of unevenly spaced vertical piles

Adviser ： Jeng-Tzong ChenDate: February 06, 2009 Place: HR2 307

Reporter : Yi-Jhou Lin

National Taiwan Ocean UniversityDepartment of Harbor and River Engineering

2

Outlines

Problem statement Numerical examples Concluding remarks

3

Problem statement

.),,(,0);,,(2 Dzyxtzyx

})(),(Re{);,,( tiezfyxtzyx

kh

hzkigAzf

cosh

)(cosh)(

Governing equation:

Separation variable :

).,(,0 yxhzn

where

Seabed boundary conditions : inc

Original Problem

inc

Free-surface conditions :

, ( , , ).z t z x y yH H H z H x y t

2 2( ) ( , ) 0, ( , )k x y x y D

4

Decompose two parts

inc

＝

Radiation field (typical BVP)

Free field

inc

Original Problem

inc inc

＋

5

FlowchartOriginal problem

Decompose two parts

Free field Radiation field

Expansion

Fourier series of boundary densities

Degenerate kemelFor fundamental solution

Collocate of the real boundary

Linear algebraic system

Calculation of the unknown Fourier

BIE for the domain point

Superposing the solution of two parts

Total field

6

Numerical examples

inc

Original Problem

inc

0.0

/ 0.8

/ 1.625293

inc

a d

kd

7

Near-traped mode (Duclos and Clement, 2004)

8

0

1 0

2 0

3 0

4 0

5 0

6 0

7 0

8 0

9 0

1 0 0

1 1 0

1 2 0

1 3 0

1 4 0

1 5 0

0.0

Near-traped mode (Present, 2009)

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Perturbation parameter

0.3 0.5 0.9

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Perturbation parameter

- 2 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0- 1 2

- 1 0

- 8

- 6

- 4

- 2

0

2

4

6

8

1 0

0.1 - 2 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

- 1 2

- 1 0

- 8

- 6

- 4

- 2

0

2

4

6

8

1 0

0.3 - 2 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0

- 1 2

- 1 0

- 8

- 6

- 4

- 2

0

2

4

6

8

1 0

0.5

0 5 1 0 1 5 2 0 2 5 3 0

- 1 0

- 5

0

5

1 0

0 5 1 0 1 5 2 0 2 5 3 0

- 1 0

- 5

0

5

1 0

0.7 0.9

No trapped mode is found

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Horizontal force versus wave number

0 0.5 1 1.5 2

0

1

2

3

4

5

6

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Concluding remarks

A general-purpose program for solving the water wave problems with arbitrary number, size and various locations of circular cylinders was developed.

We have proposed a BIEM formulation by using degenerate kernels, null-field integral equation and Fourier series in companion with adaptive observer system.

Near trapped mode is observed in this study.

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The end

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